This is the first in a series of posts about the Atiyah-Singer index theorem.

Let be finite-dimensional vector spaces (over , say), and consider the space of linear maps . To each , we can assign two numbers: the dimension of the kernel and the dimension of the cokernel . These are obviously nonconstant, and not even locally constant. However, the difference is constant in .

This was a trivial observation, but it leads to something deeper. More generally, let’s consider an operator (such as, eventually, a differential operator), on an infinite-dimensional Hilbert space. Choose separable, infinite-dimensional Hilbert spaces ; while they are abstractly isomorphic, we don’t necessarily want to choose an isomorphism between them. Consider a bounded linear operator .

Definition 1isFredholmif is “invertible up to compact operators,” i.e. there is a bounded operator such that and are compact.

In other words, if one forms the category of Hilbert spaces and bounded operators, and quotients by the ideal (in this category) of compact operators, then is invertible in the quotient category. It thus follows that adding a compact operator does not change Fredholmness: in particular, is Fredholm if and is compact.

Fredholm operators are the appropriate setting for generalizing the small bit of linear algebra I mentioned earlier. In fact,

Proposition 2A Fredholm operator has a finite-dimensional kernel and cokernel.

*Proof:* In fact, let be the kernel. Then if , we have

where is a “pseudoinverse” to as above. If we let range over the elements of of norm one, then the right-hand-side ranges over a compact set by assumption. But a locally compact Banach space is finite-dimensional, so is finite-dimensional. Taking adjoints, we can similarly see that the cokernel is finite-dimensional (because the adjoint is also Fredholm).

The space of Fredholm operators between a pair of separable, infinite-dimensional Hilbert spaces is interesting. For instance, it has the homotopy type of , so it is a representing space for K-theory. In particular, the space of its connected components is just . The stratification of the space of Fredholm operators is given by the *index*.

Definition 3Given a Fredholm operator , we define theindexof to be . (more…)